Let we have the following sequence in $L^2$ The sequence is $$X=(x_1,x_2,x_3,......,x_n,......)$$ Such that $$x_1=(1,0,0,0,0,0,,.....)$$ $$x_2=(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},0,0,0,0,,...)$$ $$x_3=(\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},0,0,0,0,.....)$$ $$............$$ $$x_n=(\frac{1}{n},\frac{1}{n},\frac{1}{n},\frac{1}{n},\frac{1}{n},.....,\frac{1}{n},0,0,0,0,,....)$$ I mean in any ($x_n$) we repeat the fraction ($\frac{1}{n}$) in the sequence ( $n^2$ )times . Now how can I study the convergence of this sequence $X$ in $L^2$ space
2026-04-13 12:01:40.1776081700
How can I study the convergence of the following sequence?
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Simple: The sequence does not converge. The only possible candidate to which it could converge is the all-$0$ sequence (because if it converges, it converges in each coordinate, and by coordinates, it converges to $0$), but the norm of the elements clearly does not converge toward $0$.